Okay, so this is a brief summary of how conics and arithmetic can give a pretty unifying look at Euler's formula and some other related ideas. The main observation and inciting question is that both circular and hyperbolic trig functions are defined via angles, that these functions have nice properties, but that the definitions of angles can be a bit ad hoc. Why should these particular definitions of angles work to make such nice functions? So we want a definition for "angle" that is a bit more natural and universal.

To do this, we are going to turn all conics into groups. More specifically, if X is a conic and O is a marked point on it, then through purely geometric means we can turn X into a group with identity O. If A and B are points on X, then we can draw the line through O that is parallel to the line through A and B, this will intersect X at a second point which we call A B. With some work, it can be proved that this is a group and, in fact, is a Lie Group.

The group structure for any conic is determined by the type of conic it is. Any ellipse is isomorphic to the group of rotations of the unit circle. Any hyperbola is isomorphic to the group of hyperbolic rotations of the unit hyperbola, which is also isomorphic to multiplication. Any parabola is isomorphic to the parabola y=x², which is effectively addition.

So every conic is a Lie Group, which means we can make sense of "infinitesimal addition". Practically, if you have a particle that is on a conic, you can give it some initial velocity and then it will draw out a path on the conic that follows the "flow" given by the group addition. For circles, for instance, it will rotate at that initial constant velocity. The set of initial velocities is called the Lie Algebra. If A is a point on the conic X, and if we can nudge a particle at O with some initial velocity v so that it reaches A in one unit of time, then we can say that v is an "angle" for A. For example, if we're on the unit circle and give a particle at point (1,0) rotational velocity x, then it will follow the path (cos(xt), sin(xt)) reaching (cos(x), sin(x)) at time t=1. So x is the angle for this point.

If L is the Lie Algebra for the conic X, then there is a function T:L->X so that T(x) is the point on X that a particle with initial velocity x will reach when starting at O when following the "flow" given by the Lie Group X. If X is the unit circle, then T(x)=(cos(x), sin(x)). If X is the unit hyperbola, then T(x)=(cosh(x), sinh(x)). If X is the hyperbola xy=1, then T(x)=(exp(x), exp(-x)). If X is the parabola y=x², then T(x)=(x, x^(2)).

The interesting thing is that given a value v in the tangent space, you can define a vector field on the entire conic defined by v, say that v(P) is the vector at the point P. If you then have any differentiable path c(t) from O to a point A, then you can write its derivative as c'(t)=g(t)v(c(t)). Now, if g(t) is constant, then the path is already a path that follows the group "flow", and then the angle for A would be that constant (or, the constant divided by the value of the v we started with). If g(t) is not constant, then we can actually average over it and compute the integral of g(t)dt for t=0 to 1. It then turns out that this average is the angle for the point A! You can recover the formulas for inverse trig, hyperbolic, and logarithmic functions following this procedure.

Something to note, then, is that the building blocks for all elementary functions arise from the group structure of conics. Which is fun. So we have a pretty unifying theory of trig functions and angles. This underlying groupiness points to some important properties of these functions. Most notably is the additive nature of all these kinds of functions. But isomorphisms between conics produce relationships between the various functions. For instance, there is an isomorphism between the unit hyperbola x²-y²=1 and the hyperbola xy=1 given by (x,y) -> (x y,x-y). This group isomorphism produces an isomorphism between their respective functions, namely the hyperbolic trig functions and exponential functions. And so it is this isomorphism that is responsible for the equation exp(x)=cosh(x) sinh(x).

Now, nothing about this situation changes if we are working with complex numbers. The only change is that the conics are complex surfaces and the angles can now be complex. This allows us to make sense of things like cos(ix) purely geometrically and without having to resort to infinite series. In fact, that the unit circle x² y²=1 becomes isomorphic to the unit hyperbola x^2-y²=1 through (x,y)->(x,iy) gives rise to formulas like cos(ix)=cosh(x).

Finally, the unit circle becomes isomorphic to the hyperbola xy=1 when we use complex numbers through the isomorphism (x,y)->(x iy,x-iy). This results in Euler's formula, exp(ix)=cos(x) isin(x) in the exact same way that we got the hyperbolic version of Euler's formula!

So this is just a fun way to look at some of the properties and formulas involving elementary functions. It is "just" Lie Groups, Lie Algebras, and homomorphisms, but the somewhat novel thing is universally defining this group structure on the conics in order to appeal to Lie theory to get these results. There are tons more details, and some of the specific examples are illuminating and satisfying, but I haven't gotten time to more formally write it down in much detail so I just wanted to document some thoughts here.